
the expected bias of a random averaged long code is \(\mu \) . To show concentration around the expectation, we shall use the following key lemma from [1]}:
[Lemma 6.7 [1]}] 
Let \(\lbrace f_A\rbrace _{A \in V^R}\)  be a set of functions \(f_A : \Omega ^R \rightarrow [0,1]\) . Furthermore, define \(g_A\)  as
\(g_A(x,z) \overset{\rm def}{=}_{B \sim G^{\otimes R}_\eta (A)} _{(B^{\prime },x^{\prime }) \sim M_z(B,x)}_{\pi \sim {S}_R} \left[f\left(\pi \circ \left(B^{\prime },x^{\prime },z\right)\right) \right],\) 
